Mathematical Notes

, Volume 78, Issue 3–4, pp 498–502 | Cite as

On Brennan's Conjecture for a Special Class of Functions

  • I. P. Kayumov


In this paper, we prove Brennan's conjecture for conformal mappings f of the disk {z : | z| < 1} assuming that the Taylor coefficients of the function log(zf′(z)/f(z)) at zero are nonnegative. We also obtain inequalities for the integral means over the circle |z| = r of the squared modulus of the function zf′(z)/f(z).

Key words

Brennan's conjecture univalent analytic function Koebe function conformal mapping fractal boundary 


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  1. 1.
    J. E. Brennan, “On the integrability of the derivative in conformal mapping,” J. London Math. Soc. (2), 18 (1978), 261–272.Google Scholar
  2. 2.
    Ch. Pommerenke, “On the integral means of the derivative of a univalent function,” J. London Math. Soc. (2), 32 (1985), 254–258.Google Scholar
  3. 3.
    L. Carleson and N. G. Makarov, “Some results connected with Brennan's conjecture,” Ark. Mat., 32 (1994), 33–62.Google Scholar
  4. 4.
    D. Bertilsson, On Brennan's conjecture in conformal mapping, Doctoral Thesis, Royal Inst. of Tech., Stockholm, 1999.Google Scholar
  5. 5.
    K. Baranski and A. Volberg, and A. Zdunik, “Brennan's conjecture and the Mandelbrot set,” Intern. Math. Res. Notices, 12 (1998), 589–600.CrossRefGoogle Scholar
  6. 6.
    I. R. Kayumov, Integral Means and the Law of the Iterated Logarithm, Preprint no. 8, Institut Mittag-Leffler, Sweden, 2002.Google Scholar
  7. 7.
    I. M. Milin, Univalent Functions and Orthonormal Systems [in Russian], Nauka, Moscow, 1971.Google Scholar
  8. 8.
    I. E. Bazilevich, “On a criterion for the univalence of regular functions and the variance of their coefficients,” Mat. Sb. [Math. USSR-Sb.], 74 (1967), no. 1, 133–146.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • I. P. Kayumov
    • 1
  1. 1.N. G. Chebotarev Research Institute of Mathematics and MechanicsKazan State UniversityKazanRussia

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